A general transformation class of semiparametric cure rate frailty models

Article

Abstract

We consider a class of cure rate frailty models for multivariate failure time data with a survival fraction. This class is formulated through a transformation on the unknown population survival function. It incorporates random effects to account for the underlying correlation, and includes the mixture cure model and the proportional hazards cure model as two special cases. We develop efficient likelihood-based estimation and inference procedures. We show that the nonparametric maximum likelihood estimators for the parameters of these models are consistent and asymptotically normal, and that the limiting variances achieve the semiparametric efficiency bounds. Simulation studies demonstrate that the proposed methods perform well in finite samples. We provide an application of the proposed methods to the data of the age at onset of alcohol dependence, from the Collaborative Study on the Genetics of Alcoholism.

Keywords

Box-Cox transformation Cure fraction Empirical process Mixture cure model NPMLE Proportional hazards cure model Semiparametric efficiency 

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References

  1. Abecasis G.R., Cardon L.R., Cookson W.O.C. (2000) A general test of association for quantitative traits in nuclear families. American Journal of Human Genetics 66: 279–292CrossRefGoogle Scholar
  2. Bailey-Wilson, J. E., Thomas, D., MacCluer, J. W. (2005). Genetic Analysis Workshop 14: Summarizing analyses comparing microsatellite and SNP marker loci for genome-wide scans. Genetic Epidemiology, 29(Suppl 1), S1–S132.Google Scholar
  3. Begleiter H., Reich T., Hesselbrock V., Porjesz B., Li T.K., Schuckit M.A., Edenberg H.J., Rice J.P. (1995) The collaborative study on the genetics of alcoholism. Alcohol Health Res World 19: 228–236Google Scholar
  4. Berkson J., Gage R.P. (1952) Survival curve for cancer patients following treatment. Journal of the American Statistical Association 47: 501–515CrossRefGoogle Scholar
  5. Betensky R.A., Schoenfeld D.A. (2001) Nonparametric estimation in a cure model with random cure times. Biometrics 57: 282–286MathSciNetMATHCrossRefGoogle Scholar
  6. Bickel P.J., Klaassen C.A.J., Ritov Y., Wellner J.A. (1993) Efficient and adaptive estimation for semiparametric models. Johns Hopkins University Press, BaltimoreMATHGoogle Scholar
  7. Box G.E.P., Cox D.R. (1964) An analysis of transformations (with discussion). Journal of the Royal Statistical Society. Series B 26: 211–252MathSciNetMATHGoogle Scholar
  8. Chatterjee N., Shih J. (2001) A bivariate cure-mixture approach for modeling familial association in diseases. Biometrics 57: 779–786MathSciNetMATHCrossRefGoogle Scholar
  9. Chen M.H., Ibrahim J.G., Sinha D. (1999) A new Bayesian model for survival data with a surviving fraction. Journal of the American Statistical Association 94: 909–919MathSciNetMATHCrossRefGoogle Scholar
  10. Chen M.H., Ibrahim J.G., Sinha D. (2002) Bayesian inference for multivariate survival data with a cure fraction. Journal of Multivariate Analysis 80: 101–126MathSciNetMATHCrossRefGoogle Scholar
  11. Clayton D.G. (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65: 141–151MathSciNetMATHCrossRefGoogle Scholar
  12. Cooner F., Banerjee S., Carlin B.P., Sinha D. (2007) Flexible cure rate modeling under latent activation schemes. Journal of the American Statistical Association 102: 560–572MathSciNetMATHCrossRefGoogle Scholar
  13. Cox D.R. (1972) Regression models and life-tables (with discussion). Journal of the Royal Statistical Society, Series B 34: 187–220MATHGoogle Scholar
  14. Diao G., Lin D.Y. (2006) Semiparametric variance-component models for linkage and association analyses of censored trait data. Genetic Epidemiology 30: 570–581CrossRefGoogle Scholar
  15. Glidden D.V. (2007) Pairwise dependence diagnostics for clustered failure-time data. Biometrika 94: 371–385MathSciNetMATHCrossRefGoogle Scholar
  16. Gray R.J., Tsiatis A.A. (1989) A linear rank test for use when the main interest is in differences in cure rates. Biometrics 45: 899–904MATHCrossRefGoogle Scholar
  17. Kuk A.Y.C., Chen C.H. (1992) A mixture model combining logistic regression with proportional hazards regression. Biometrika 79: 531–541MATHCrossRefGoogle Scholar
  18. Li C.S., Taylor J.M.G., Judy P.S. (2001) Identifiability of cure models. Statistics & Probability Letters 54: 389–395MathSciNetMATHCrossRefGoogle Scholar
  19. Locatelli I., Rosina A., Lichtenstein P., Yashin A. (2007) A correlated frailty model with long-term survivors for estimating the heritability of breast cancer. Statistics in Medicine 26: 3722–3734MathSciNetCrossRefGoogle Scholar
  20. Maller R., Zhou X. (1996) Survival analysis with long-term survivors. Wiley, New YorkMATHGoogle Scholar
  21. Murphy S.A., van der Vaart A.W. (2000) On profile likelihood. Journal of the American Statistical Association 95: 449–465MathSciNetMATHCrossRefGoogle Scholar
  22. Peng Y., Dear K.B.G. (2000) A nonparametric mixture model for cure rate estimation. Biometrics 56: 237–243MATHCrossRefGoogle Scholar
  23. Peng Y., Taylor J.M.G., Yu B. (2007) A marginal regression model for multivariate failure time data with a surviving fraction. Lifetime Data Analysis 13: 351–369MathSciNetMATHCrossRefGoogle Scholar
  24. Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P. (1992). Numerical Recipes in C: The art of scientific computing (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
  25. Price D.L., Manatunga A.K. (2001) Modelling survival data with a cured fraction using frailty models. Statistics in Medicine 20: 1515–1527CrossRefGoogle Scholar
  26. Rudin W. (1973) Functional analysis. McGraw-Hill, New YorkMATHGoogle Scholar
  27. Sy J.P., Taylor J.M.G. (2000) Estimation in a Cox proportional hazards cure model. Biometrics 56: 227–236MathSciNetMATHCrossRefGoogle Scholar
  28. Taylor J.M.G. (1995) Semi-parametric estimation in failure time mixture models. Biometrics 51: 899–907MATHCrossRefGoogle Scholar
  29. Tsodikov A.D. (1998) A proportional hazards model taking account of long-term survivors. Biometrics 54: 1508–1516MATHCrossRefGoogle Scholar
  30. Tsodikov A.D., Ibrahim J., Yakovlev A.Y. (2003) Estimating cure rates from survival data: An alternative to two-component mixture models. Journal of the American Statistical Association 98: 1063–1078MathSciNetCrossRefGoogle Scholar
  31. van der Vaart A.W., Wellner J.A. (1996) Weak convergence and empirical processes. Springer, New YorkMATHGoogle Scholar
  32. Yakovlev, A. Y., Asselain, B., Bardou, V. J., Fourquet, A., Hoang, T., Rochefediere, A., Tsodikov, A. D. (1993). A simple stochastic model of tumour recurrence and its applications to data on premenopausal breast cancer. In B. Asselain, M. Boniface, C. Duby, C. Lopez, J. P. Masson, J. Tranchefort (Ed.), Biometrie et Analyse de Dormees Spatio-Temporelles (Vol. 12, pp. 66–82). France: Société Francaise de Biométrie, ENSA Renned.Google Scholar
  33. Yau K.K.W., Ng A.S.K. (2001) Long-term survivor mixture model with random effects: Application to a multicentre clinical trial of carcinoma. Statistics in Medicine 20: 1591–1607CrossRefGoogle Scholar
  34. Yin G. (2008) Bayesian transformation cure frailty models with multivariate failure time data. Statistics in Medicine 27: 5929–5940MathSciNetCrossRefGoogle Scholar
  35. Yin G., Ibrahim J. (2005) Cure rate models: A unified approach. The Canadian Journal of Statistics 33: 559–570MathSciNetMATHCrossRefGoogle Scholar
  36. Yu B., Peng Y. (2008) Mixture cure models for multivariate survival data. Computational Statistics and Data Analysis 52: 1524–1532MathSciNetMATHCrossRefGoogle Scholar
  37. Zeng D., Lin D.Y. (2007) Maximum likelihood estimation in semiparametric regression models with censored data (with discussion). Journal of the Royal Statistical Society: Series B 69: 507–564MathSciNetCrossRefGoogle Scholar
  38. Zeng D., Lin D.Y., Lin X. (2008) Semiparametric transformation models with random effects for clustered failure time data. Statistica Sinica 18: 355–377MathSciNetMATHGoogle Scholar
  39. Zeng D., Yin G., Ibrahim J. (2006) Semiparametric transformation models for survival data with a cure fraction. Journal of American Statistical Association 101: 670–684MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2012

Authors and Affiliations

  1. 1.Department of StatisticsGeorge Mason University MS 4A7FairfaxUSA
  2. 2.Department of Statistics and Actuarial ScienceUniversity of Hong KongHong KongChina

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